scholarly journals Diophantine problems in solvable groups

2020 ◽  
Vol 10 (01) ◽  
pp. 2050005 ◽  
Author(s):  
Albert Garreta ◽  
Alexei Miasnikov ◽  
Denis Ovchinnikov
Keyword(s):  
Number Theory ◽  
Polynomial Time ◽  
Solvable Groups ◽  
Systems Of Equations ◽  
Heisenberg Groups ◽  
Finitely Generated ◽  
Diophantine Problem ◽  
Ring Of Integers ◽  
Finite Systems ◽  
Ring Of Algebraic Integers

We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc.), which satisfy some natural “non-commutativity” conditions. For each group [Formula: see text] in one of these classes, we prove that there exists a ring of algebraic integers [Formula: see text] that is interpretable in [Formula: see text] by finite systems of equations ([Formula: see text]-interpretable), and hence that the Diophantine problem in [Formula: see text] is polynomial time reducible to the Diophantine problem in [Formula: see text]. One of the major open conjectures in number theory states that the Diophantine problem in any such [Formula: see text] is undecidable. If true this would imply that the Diophantine problem in any such [Formula: see text] is also undecidable. Furthermore, we show that for many particular groups [Formula: see text] as above, the ring [Formula: see text] is isomorphic to the ring of integers [Formula: see text], so the Diophantine problem in [Formula: see text] is, indeed, undecidable. This holds, in particular, for free nilpotent or free solvable non-abelian groups, as well as for non-abelian generalized Heisenberg groups and uni-triangular groups [Formula: see text]. Then, we apply these results to non-solvable groups that contain non-virtually abelian maximal finitely generated nilpotent subgroups. For instance, we show that the Diophantine problem is undecidable in the groups [Formula: see text].

scholarly journals Systems of equations over finitely generated solvable groups

1989 ◽  
Vol 33 (2) ◽  
pp. 244-249
Author(s):  
Wojtek Jastrzebowski ◽  
John Lawrence
Keyword(s):  
Solvable Groups ◽  
Systems Of Equations ◽  
Finitely Generated

Parametrizing Fuchsian Subgroups of the Bianchi Groups

1991 ◽  
Vol 43 (1) ◽  
pp. 158-181 ◽  
Author(s):  
C. Maclachlan ◽  
A. W. Reid
Keyword(s):  
Number Theory ◽  
Group Theory ◽  
Present Article ◽  
Ring Of Integers ◽  
Low Dimensional Topology ◽  
Low Dimensional

Let dbe a positive square-free integer and let Od denote the ring of integers in . The groups PSL2(Od) are collectively known as the Bianchi groups and have been widely studied from the viewpoints of group theory, number theory and low-dimensional topology. The interest of the present article is in geometric Fuchsian subgroups of the groups PSL2(Od). Clearly PSL2 is such a subgroup; however results of [18], [19] show that the Bianchi groups are rich in Fuchsian subgroups.

On the profinite topology on solvable groups

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Khadijeh Alibabaei
Keyword(s):  
Abelian Group ◽  
Wreath Product ◽  
Solvable Group ◽  
Solvable Groups ◽  
Polycyclic Group ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Finitely Generated Group ◽  
Hnn Extension ◽  
Profinite Topology

AbstractWe show that the wreath product of a finitely generated abelian group with a polycyclic group is a LERF group. This theorem yields as a corollary that finitely generated free metabelian groups are LERF, a result due to Coulbois. We also show that a free solvable group of class 3 and rank at least 2 does not contain a strictly ascending HNN-extension of a finitely generated group. Since such groups are known not to be LERF, this settles, in the negative, a question of J. O. Button.

Describing Groups

2007 ◽  
Vol 13 (3) ◽  
pp. 305-339 ◽  
Author(s):  
André Nies
Keyword(s):  
Finite Automaton ◽  
Finite Automata ◽  
Finite Alphabet ◽  
Prime Model ◽  
Finitely Generated ◽  
Ring Of Integers ◽  
Group Operation ◽  
First Order

AbstractTwo ways of describing a group are considered. 1. A group is finite-automaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g. group is quasi-finitely axiomatizable if there is a description consisting of a single first-order sentence, together with the information that the group is finitely generated. In the first part of the paper we survey examples of FA-presentable groups, but also discuss theorems restricting this class. In the second part, we give examples of quasi-finitely axiomatizable groups, consider the algebraic content of the notion, and compare it to the notion of a group which is a prime model. We also show that if a structure is bi-interpretable in parameters with the ring of integers, then it is prime and quasi-finitely axiomatizable.

scholarly journals ON THE COMPLEXITY OF THE WHITEHEAD MINIMIZATION PROBLEM

2007 ◽  
Vol 17 (08) ◽  
pp. 1611-1634 ◽  
Author(s):  
ABDÓ ROIG ◽  
ENRIC VENTURA ◽  
PASCAL WEIL
Keyword(s):  
Free Group ◽  
Polynomial Time ◽  
Minimization Problem ◽  
Finite Rank ◽  
Polynomial Algorithm ◽  
Minimum Size ◽  
Input Word ◽  
Finitely Generated ◽  
Factor Problem

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem — to decide whether a word is an element of some basis of the free group — and the free factor problem can also be solved in polynomial time.

scholarly journals Palindromic Width of Finitely Generated Solvable Groups

2015 ◽  
Vol 43 (11) ◽  
pp. 4809-4824 ◽  
Author(s):  
Valeriy G. Bardakov ◽  
Krishnendu Gongopadhyay
Keyword(s):  
Solvable Groups ◽  
Finitely Generated
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